The mathematical theory of behavioral swarms: Toward modeling the collective dynamics of living systems

Classical swarm models, exemplified by the Cucker–Smale framework, provide foundational insights into collective alignment but exhibit fundamental limitations in capturing the adaptive, heterogeneous behaviors intrinsic to living systems. This paper formalizes the mathematical theory of Behavioral Swarms, a comprehensive framework where each particle’s state incorporates a dynamic internal variable, the activity that co-evolves with position and velocity through nonlocal interactions. We demonstrate how this approach transcends prior models by integrating adaptive decision-making mechanisms and heterogeneous behavioral states into rigorous differential systems. Through applications in behavioral economics and crowd dynamics, we establish the theory’s capacity to predict emergent macroscopic patterns from individual behavioral states. Our critical analysis positions this framework against kinetic theories of active particles and agent-based approaches, revealing distinct advantages for modeling systems where individual agency drives collective outcomes.